Hesselink strata in small characteristic and Lusztig-Xue pieces

Let G be a simple algebraic group over an algebraically closed field of characteristic p≥ 0 and g= Lie(G). We show that the nilpotent pieces LX(Δ) introduced by Lusztig form a partition of the nilpotent cone of g and hence coincide with the corresponding Hesselink strata H(Δ). Here Δ runs through the set of all weighted Dynkin diagrams of G. Thanks to the results obtained by Lusztig, Xue and Voggesberger this boils down to establishing the partition property of the pieces LX(Δ) for groups of type E₇ in characteristic $2$ and for groups of type E₈ in characteristic $2$ and $3$.